Let $X_1$, ..., $X_n$ be independent, $Exp(1)$ distributed random variables, where $X_{(0)}=c \in \mathbb{R}$ ist fixed. Define $Z_i := (n-i+1)(X_{(i)}-X_{(i-1)})$.
What is the distribution of $\sum_{i=1}^n Z_i$?
I tried to solve this problem in two steps. First, I looked at the distribution of $Z_1$, ..., $Z_n$ and via transformation could (making use of a hint from the lecture) show that they are exponentially distributed and independent again. Their density is $$f_{Z_1, ..., Z_n}(z_1, ..., z_n)=\exp(\sum_{i=1}^n-z_i).$$
I know that I should be able to conclude the distrubtion of $\sum_{i=1}^n Z_i$ from here. But I don't see how.
I have had problems with this general problem pattern before: Given the joint density of random variables $Y_1$, ..., $Y_n$, how do I calculate e.g. the density of some real valued mesaruable function $g(Y_1, ..., Y_n)$?
I was able to do it for the $Z_i$ because we've had it for the range $R=X_{(i)}-X_{(i-1)}$ in the uniform case, but I'm kind of missing the big picture here.
Can anyone help me with this?
The density of $(Z_1,...,Z_n)$ is the product of the densities of each $Z_i$, therefore they are independent. And a sum of independent exponential random variables with the same parameter is Gamma distributed.